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    General

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    Commutative algebra fall 2026


    First meeting (tentative, might change):     Monday September 7 2026 at 13.00 in room Cramér. Time and place for the remaining lectures will be decided after the first meeting.

    Teachers: Luís Duarte and Samuel Lundqvist

    Prerequisites: Knowledge of basic concepts in Abstract algebra such as ideals, rings, and modules. 

    Content: The goal of this course is to provide an introduction to some key topics in Commutative Algebra, with focus on homological methods.

    We will begin by covering some basic fundamental concepts such as localization, Noetherian rings, Artinian rings and associated primes and we will arrive at the concepts of height and dimension. We will also cover some aspects of graded rings.  

    We will then move on to the study of projective and injective resolutions of modules, which are the basis of Homological Commutative Algebra.

    We will use homological properties to characterize regular sequences, the invariant depth and the property of a ring being regular.

    For the final stretch of the course, we will delve into the study of Cohen-Macaulay rings and Gorensteins rings, and the various ways one can characterize them.

    We will closely follow some chapters of the Matsumura's and Bruns & Herzog's books, but it is possible to follow the course just using the lecture notes.

    (1) Winfried Bruns and Jürgen Herzog, "Cohen-Macaulay rings", volume 39. Cambridge University Press, 1998.
    (2) Hideyuki Matsumura, "Commutative ring theory", volume 8, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1986.

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